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Zhu, Yuming; Liu, Gusheng; Du, Feng

Eigenvalue inequalities of elliptic operators in weighted divergence form on smooth metric measure spaces. (English) Zbl 1342.35194

J. Inequal. Appl. 2016, Paper No. 191, 15 p. (2016).
Summary: In this paper, we study the eigenvalue problem of elliptic operators in weighted divergence form on smooth metric measure spaces. First of all, we give a general inequality for eigenvalues of the eigenvalue problem of elliptic operators in weighted divergence form on compact smooth metric measure space with boundary (possibly empty). Then applying this general inequality, we get some universal inequalities of Payne-Pólya-Weinberger-Yang type for the eigenvalues of elliptic operators in weighted divergence form on a connected bounded domain in the smooth metric measure spaces, the Gaussian shrinking solitons, and the general product solitons, respectively.

Cited in 1 Document

MSC:

35P15 Estimates of eigenvalues in context of PDEs
53C20 Global Riemannian geometry, including pinching
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

eigenvalue; universal inequalities; drifting Laplacian; elliptic operators in weighted divergence form; Gaussian shrinking soliton; general product soliton
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